Content uploaded by Anna Pandolfi
Author content
All content in this area was uploaded by Anna Pandolfi
Content may be subject to copyright.
A preview of the PDF is not available
Three-Dimensional Modeling and Computational Analysis of the Human Cornea Considering Distributed Collagen Fibril Orientations
Abstract and Figures
Experimental tests on human corneas reveal distinguished reinforcing collagen lamellar structures that may be well described by a structural constitutive model considering distributed collagen fibril orientations along the superior-inferior and the nasal-temporal meridians. A proper interplay between the material structure and the geometry guarantees the refractive function and defines the refractive properties of the cornea. We propose a three-dimensional computational model for the human cornea that is able to provide the refractive power by analyzing the structural mechanical response with the nonlinear regime and the effect the intraocular pressure has. For an assigned unloaded geometry we show how the distribution of the von Mises stress at the top surface of the cornea and through the corneal thickness and the refractive power depend on the material properties and the fibril dispersion. We conclude that a model for the human cornea must not disregard the peculiar collagen fibrillar structure, which equips the cornea with the unique biophysical, mechanical, and optical properties.
Figures - uploaded by Anna Pandolfi
Author content
All figure content in this area was uploaded by Anna Pandolfi
Content may be subject to copyright.
... In human corneas, the stroma is characterized by the presence of collagen fibers, which show a preferred orientation, being orthogonally arranged along the nasal-temporal (N-T) and inferior-superior (I-S) directions within the central region and circumferentially oriented in the limbal region (Fig. 4a). [36][37][38][39][40] Because the fibers are not perfectly aligned, inplane and out-of-plane dispersion terms must be considered. The degree of out-of-plane dispersion varies in depth, meaning that the fibers are more aligned with N-T and I-S directions within the two posterior thirds and are more isotropically oriented within the anterior third. ...
... To model the anisotropic behavior of the collagen fibers (Fig. 4b), the hyperelastic Holzapfel-Gasser-Odgen model with dispersion parameters was used: 36,38 ...
... in literature. 36 We chose to change only the constant C10, because it controls the extracellular matrix behavior, which mostly affects corneal tissue's mechanical response. The results of the objective refraction for the four patient-specific models are shown in Table 7. ...
Purpose:
Computational models can help clinicians plan surgeries by accounting for factors such as mechanical imbalances or testing different surgical techniques beforehand. Different levels of modeling complexity are found in the literature, and it is still not clear what aspects should be included to obtain accurate results in finite-element (FE) corneal models. This work presents a methodology to narrow down minimal requirements of modeling features to report clinical data for a refractive intervention such as PRK.
Methods:
A pipeline to create FE models of a refractive surgery is presented: It tests different geometries, boundary conditions, loading, and mesh size on the optomechanical simulation output. The mechanical model for the corneal tissue accounts for the collagen fiber distribution in human corneas. Both mechanical and optical outcome are analyzed for the different models. Finally, the methodology is applied to five patient-specific models to ensure accuracy.
Results:
To simulate the postsurgical corneal optomechanics, our results suggest that the most precise outcome is obtained with patient-specific models with a 100 µm mesh size, sliding boundary condition at the limbus, and intraocular pressure enforced as a distributed load.
Conclusions:
A methodology for laser surgery simulation has been developed that is able to reproduce the optical target of the laser intervention while also analyzing the mechanical outcome.
Translational relevance:
The lack of standardization in modeling refractive interventions leads to different simulation strategies, making difficult to compare them against other publications. This work establishes the standardization guidelines to be followed when performing optomechanical simulations of refractive interventions.
... The geometry of the developed finite element model, based on the corneal topography, is affected by the IOP that loads the tissue. The stress-strain distributions in the physiological state are unknown and must be calculated by considering the stress-free geometry (SFG) that is calculated with iterative methods [12,23]. These processes are time-consuming and affect the use of in silico models in real-time clinical applications. ...
... By considering the new anterior and posterior corneal surfaces and their corneal thickness, a 3D corneal geometric model was generated. These patient-specific geometrical models have been used to diagnose keratoconus disease [23,31,32]. ...
Implementing in silico corneal biomechanical models for surgery applications can be boosted by developing patient-specific finite element models adapted to clinical requirements and optimized to reduce computational times. This research proposes a novel corneal multizone-based finite element model with octants and circumferential zones of clinical interest for material definition. The proposed model was applied to four patient-specific physiological geometries of keratoconus-affected corneas. Free-stress geometries were calculated by two iterative methods, the displacements and prestress methods, and the influence of two boundary conditions: embedded and pivoting. The results showed that the displacements, stress and strain fields differed for the stress-free geometry but were similar and strongly depended on the boundary conditions for the estimated physiological geometry when considering both iterative methods. The comparison between the embedded and pivoting boundary conditions showed bigger differences in the posterior limbus zone, which remained closer in the central zone. The computational calculation times for the stress-free geometries were evaluated. The results revealed that the computational time was prolonged with disease severity, and the displacements method was faster in all the analyzed cases. Computational times can be reduced with multicore parallel calculation, which offers the possibility of applying patient-specific finite element models in clinical applications.
... 8 Other numerical models in this field include the biomechanics of the human cornea, 9 and a 3D fluid-solid interaction model of the air-puff test in the human cornea. 10 Additional computational models in several areas of ophthalmology have been presented [11][12][13][14] as well as in the references therein. ...
Purpose: This study presents a novel application of the semianalytical residual power series method to investigate a one-dimensional fractional anisotropic curvature equation describing the human cornea, the outermost layer of the eye. The fractional boundary value problem, involving the fractional derivative of curvature, poses challenges that conventional methods struggle to address. Methods: The analytical results are obtained by utilizing the simple and efficient residual power series method. The proposed method is accessible to researchers in all medical fields and is extendable to various models in disease spread and control. Results: The derived solution is a crucial outcome of this study. Through the application of the proposed method to the corneal shape model, an explicit formula for the curvature profile is obtained. To validate the solution, direct comparisons are made with numerical solutions for the integer case and other analytical solutions available in the literature for the fractional case. Conclusion: Our findings highlight the potential of the proposed method to significantly contribute to the diagnosis and treatment of various ophthalmological conditions.
... Average mechanical properties of human corneal tissue can be estimated ex vivo using simple experimental tests [6,36], but more recent work has highlighted how material properties vary through the tissue thickness [27,40,38]. Aiming to understand how the organization of the constituents of the cornea affects the overall mechanical response, theoretical and numerical approaches have 1 been used to introduce this information in constitutive models [7,22,34,26,39]. In general, the human cornea is modelled as nearly-incompressible hyperelastic material with highly nonlinear behaviour. ...
We introduce a discrete mathematical model for the mechanical behaviour of a planar slice of human corneal tissue, in equilibrium under the action of physiological intraocular pressure (IOP). The model considers a regular (two-dimensional) network of structural elements mimicking a discrete number of parallel collagen lamellae connected by proteoglycan-based chemical bonds (crosslinks). Since the thickness of each collagen lamella is small compared to the overall corneal thickness, we upscale the discrete force balance into a continuum system of partial differential equations and deduce the corresponding macro-scopic stress tensor and strain energy function for the micro-structured corneal tissue. We demonstrate that, for physiological values of the IOP, the predictions of the discrete model converge to those of the continuum model. We use the continuum model to simulate the progression of the degenerative disease known as keratoconus, characterized by a localized bulging of the corneal shell. We assign a spatial distribution of damage (i. e., reduction of the stiffness) to the mechanical properties of the structural elements and predict the resulting macroscopic shape of the cornea, showing that a large reduction in the element stiffness results in substantial corneal thinning and a significant increase in the curvature of both the anterior and posterior surfaces.
We setup a mechanically based finite element model to evaluate the change in the shape of the human cornea induced by ablation of stromal tissue. By consid- ering the deformability of the cornea, the model computes the change of the dioptric power resulting from ablative laser surgery. We use a previously developed 3-D finite element model of the human cornea (Pandolfi and Man- ganiello in Biomech Model Mechanobiol 5:237-246, 2006). The solid geometry is discretized into finite ele- ments by an automatic procedure which recovers the unloaded configuration. The geometry is defined in para- metric form and can be characterized by individual geometrical data when available. A two-fiber reinforced hyperelastic material model, which accounts for the orga- nization of the anisotropic collagen structure, is adopted to describe the stromal tissue. For the simulation of laser refractive surgery of myopic and astigmatic eyes, a geo- metrical correction of the corneal profile is included into the code. We show two examples of application of the model to the reshaping of a myopic and an astigmatic eye. Numerical results provide the postoperative shape of the cornea, the corrected refractive power, and the distribution of the stress throughout the stromal tissue.
Numerical modelling based on finite element analysis is used to represent the biomechanical behaviour of the cornea. The construction details of the model including the discretisation method, the mesh density, the thickness distribution, the topography idealisation, the boundary conditions and the material properties, are optimised to improve efficiency. Factors which are found to have a considerable effect on model accuracy are considered and those with effect below a certain low threshold are ignored to reduce cost of analysis. The model is validated against laboratory tests involving pressure inflation of corneal trephinates while monitoring their behaviour. To illustrate the potential of the validated model in studying corneal biomechanics, its use in modelling Goldmann applanation tonometry (GAT) is briefly described. In studying GAT, the model is able to accurately trace the behaviour of the cornea under tonometric pressure and monitor the gap closure and the progress of deformation to the point of applanation.
The cornea and sclera together form the tough outer tunic of the eye, which withstands the intra-ocular pressure from within and protects the contents from mechanical injury from without. The cornea is the major component in the optical system of the eye; of the total dioptric power of the human eyeball, nearly three-quarters is contributed by the interface between the cornea and the air. To this end, it is curved and transparent, and its surfaces, particularly the external one, are smooth to good optical standards. Generally, the globe of the mammalian eye approximates to a sphere, sometimes rather flattened in the antero-posterior direction, so that the curvature of cornea and sclera is similar. Its radius ranges from 1.75 mm in the mouse to 25 mm in the horse, among the more common animals. In man, and to a lesser extent in many other species, the cornea is more curved than the eyeball as a whole.
Clinical studies have identified factors such as the stent design and the deployment technique that are one cause for the success or failure of angioplasty treatments. In addition, the success rate may also depend on the stenosis type. Hence, for a particular stenotic artery, the optimal intervention can only be identified by studying the influence of factors such as stent type, strut thickness, geometry of the stent cell, and stent–artery radial mismatch with the wall. We propose a methodology that allows a set of stent parameters to be varied, with the aim of evaluating the difference in the mechanical environment within the wall before and after angioplasty with stenting. Novel scalar quantities attempt to characterize the wall changes in form of the contact pressure caused by the stent struts, and the stresses within the individual components of the wall caused by the stent. These quantities are derived numerically and serve as indicators, which allow the determination of the correct size and type of the stent for each individual stenosis. In addition, the luminal change due to angioplasty may be computed as well. The methodology is demonstrated by using a full three-dimensional geometrical model of a postmortem specimen of a human iliac artery with a stenosis using imaging data. To describe the material behavior of the artery, we considered mechanical data of eight different vascular tissues, which formed the stenosis. The constitutive models for the tissue components capture the typical anisotropic, nonlinear and dissipative characteristics under supra-physiological loading conditions. Three-dimensional stent models were parametrized in such a way as to enable new designs to be generated simply with regard to variations in their geometric structure. For the three-dimensional stent–artery interaction we use a contact algorithm based on smooth contact surfaces of at least C1-continuity, which prevents numerical problems known from standard facet-based contact algorithm. The proposed methodology has the potential to provide a scientific basis for optimizing treatment procedures and stent geometries and materials, to help stent designers examine new stent designs “virtually,” and to assist clinicians in choosing the most suitable stent for a particular stenosis.
The aim of refractive corneal surgery is to modify the curvature of the cornea to improve its dioptric properties. With that goal, the surgeon has to define the appropriate values of the surgical parameters in order to get the best clinical results, i.e., laser and geometric parameters such as depth and location of the incision, for each specific patient. A biomechanical study before surgery is therefore very convenient to assess quantitatively the effect of each parameter on the optical outcome. A mechanical model of the human cornea is here proposed and implemented under a finite element context to simulate the effects of some usual surgical procedures, such as photorefractive keratectomy (PRK), and limbal relaxing incisions (LRI). This model considers a nonlinear anisotropic hyperelastic behavior of the cornea that strongly depends on the physiological collagen fibril distribution. We evaluate the effect of the incision variables on the change of curvature of the cornea to correct myopia and astigmatism. The obtained results provided reasonable and useful information in the procedures analyzed. We can conclude from those results that this model reasonably approximates the corneal response to increasing pressure. We also show that tonometry measures of the IOP, underpredicts its actual value after PRK or LASIK surgery.
In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined.The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.
This paper presents a viscoelastic model for the fully three-dimensional stress and deformation response of fiber-reinforced composites that experience finite strains. The composites are thought to be (soft) matrix materials which are reinforced by two families of fibers so that the mechanical properties of the composites depend on two fiber directions. The relaxation and/or creep response of each compound of the composite is modeled separately and the global response is obtained by an assembly of all contributions. We develop novel closed-form expressions for the fourth-order elasticity tensor (tangent moduli) in full generality. Constitutive models for orthotropic, transversely isotropic and isotropic hyperelastic materials at finite strains with or without dissipation are included as special cases. In order to clearly show the good performance of the constitutive model, we present 3D and 2D numerical simulations of a pressurized laminated circular tube which shows an interesting `stretch inversion phenomenon' in the low pressure domain. Numerical results are in good qualitative agreement with experimental data and approximate the observed strongly anisotropic physical response with satisfying accuracy. A third numerical example is designed to illustrate the anisotropic stretching process of a fiber-reinforced rubber bar and the subsequent relaxation behavior at finite strains. The material parameters are chosen so that thermodynamic equilibrium is associated with the known homogeneous deformation state.